Quadratic EquationHard
Question
How many roots does the equation $x^{3} + ax + 2 = 0$ possess depending on ' $a$ '?
Options
A.1 if $a \in ( - \infty, - 3)$
B.3 if $a \in ( - \infty, - 3)$
C.2 if $a = - 3$
D.1 if $a \in ( - 3,\infty)$
Solution
$f(x) = \frac{x^{3} + 2}{x} = x^{2} + \frac{2}{x} = - a$
Number of roots $= \left\{ \begin{matrix} 1 & \text{~if~} & - a < 3 \Rightarrow a \in ( - 3,\infty) \\ 2 & \text{~if~} & a = - 3 \\ 3 & \text{~if~} & - a > 3 \Rightarrow a \in ( - \infty, - 3) \end{matrix} \right.\ $
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